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Full text of "Analytical Mechanics"

PERIODIC MOTION
311
Making these substitutions In the left-hand member of the oreceding expression for the period we obtain
= 4
= 4
I ffl
mgD^Q L
[by (XII)]
[when a, is small higher! L terms may be neglected J
when a is small sin^ = ~ 
Therefore
(XIV)
242.    Simple   Pendulum.  A  ball
which is suspended by means of a
string forms a simple pendulum when
it is free to swing about a horizontal
axis through the upper end of the
string, provided the mass of the string
is negligible compared with that of the    /
ball and the radius of the ball is negli- <^
gible compared with the length of the
string.   If m denotes the mass of the
ball and I the length of the string then            -pIG 138   >rmg
* This is called an elliptic integral.
t This expansion is carried out by the Binomial theorem.    See Appendix A-
J See Appendix Ai.